[Paper Review] The Capacity of Symmetric Private Information Retrieval under Arbitrary Collusion and Eavesdropping Patterns

Jiale Cheng, Nan Liu+1 more|arXiv (Cornell University)|Oct 16, 2020

Cryptography and Data Security4 citations

TL;DR

This paper establishes the capacity of symmetric private information retrieval (SPIR) under arbitrary collusion and eavesdropping patterns in replicated databases. It shows that the capacity depends on a derived parameter $ F^* $, the optimal solution to a linear program based on the union of collusion and eavesdropping patterns, revealing that collusion and eavesdropping constraints are interchangeable in terms of capacity impact.

ABSTRACT

We study the symmetric private information retrieval (SPIR) problem under arbitrary collusion and eavesdropping patterns for replicated databases. We find its capacity, which is the same as the capacity of the original SPIR problem with the number of databases $N$ replaced by a number $F^*$. The number $F^*$ is the optimal solution to a linear programming problem that is a function of the joint pattern, which is the union of the collusion and eavesdropping pattern. This is the first result that shows how two arbitrary patterns collectively affect the capacity of the PIR problem. We draw the conclusion that for SPIR problems, the collusion and eavesdropping constraints are interchangeable in terms of capacity. As special cases of our result, the capacity of the SPIR problem under arbitrary collusion patterns and the capacity of the PIR problem under arbitrary eavesdropping patterns are also found.

Motivation & Objective

• To determine the capacity of symmetric private information retrieval (SPIR) when servers may collude and external eavesdroppers may monitor queries and responses.
• To model the joint effect of arbitrary collusion and eavesdropping patterns on SPIR capacity, moving beyond symmetric or uniform patterns.
• To derive a unified capacity expression that generalizes prior results for T-colluding and T-ESPIR settings.
• To prove that collusion and eavesdropping constraints are interchangeable in their impact on capacity, meaning their combined effect depends only on their union.

Proposed method

• Formalizes the SPIR problem with arbitrary collusion and eavesdropping patterns, defining joint pattern $ ilde{ ho} $ as the union of collusion and eavesdropping sets.
• Introduces a linear programming formulation to compute $ F^* $, the effective number of independent databases under the joint pattern.
• Applies standard information-theoretic converse techniques to derive lower bounds on common randomness $ H(S) $ and download cost $ H(A_{[1:N]}^{[k]}) $.
• Uses the MDS property of coding matrices to ensure privacy against passive eavesdroppers observing up to $ E $ servers.
• Derives a lower bound on the ratio $ \rho = H(S)/H(W_k) $, leading to an upper bound on capacity $ C \leq 1 - 1/F^* $.
• Proves that the capacity is exactly $ 1 - 1/F^* $, where $ F^* $ is the optimal solution to the LP derived from the joint pattern.

Experimental results

Research questions

• RQ1 How does an arbitrary combination of collusion and eavesdropping patterns affect the capacity of symmetric private information retrieval?
• RQ2 Can the joint impact of collusion and eavesdropping be captured by a single effective parameter in the capacity formula?
• RQ3 Is the effect of collusion on capacity equivalent to that of eavesdropping under the same pattern constraints?
• RQ4 Does the capacity of SPIR depend only on the union of collusion and eavesdropping sets, rather than their individual structures?
• RQ5 Can the capacity formula for standard SPIR be generalized to arbitrary patterns using a unified optimization framework?

Key findings

• The capacity of SPIR under arbitrary collusion and eavesdropping patterns is $ C = 1 - 1/F^* $, where $ F^* $ is the optimal solution to a linear program based on the joint pattern.
• The joint pattern, defined as the union of all collusion and eavesdropping sets, fully determines the capacity, making the two constraints interchangeable in their effect.
• The result generalizes prior capacity formulas: for $ T $-colluding SPIR, $ F^* = N/T $; for $ E $-eavesdropping SPIR, $ F^* = N/E $; and for $ T $-ESPIR, $ F^* = N/\max(T,E) $.
• The capacity is independent of the number of messages $ K $, consistent with the original SPIR result.
• The lower bound on common randomness $ \rho \geq 1/(F^* - 1) $ is tight, confirming the optimality of the derived capacity.
• The proof technique extends standard SPIR converse arguments by incorporating both colluding and eavesdropping sets into a unified inequality framework using weighted sums over pattern sets.